Two Stream Plasma Instability
Defining the Equations of Motion
Linearizing the Equations
Solving for the Dispersion Relation
.latex
\begin{pmatrix}
V_0 k - \omega & 0 & n_0 k & 0 & 0 \\
0 & 0 & i m_e(V_0 k - \omega) & 0 & e \\
0 & - \omega & 0 & n_0 k & 0 \\
0 & 0 & 0 & i m_j \omega & e \\
1 & -1 & 0 & 0 & (\epsilon_0 i k) / e
\end{pmatrix}
\begin{pmatrix}
\tilde{n}_{e1} \\ \tilde{n}_{j1} \\ \tilde{v}_{e1} \\ \tilde{v}_{j1} \\ \tilde{E}_1
\end{pmatrix} = 0 \quad \iff \quad M \mathbf{u} = 0
(*Define our variables*)
\[Epsilon]0 = Subscript[\[Epsilon], 0]; me = Subscript[m, e]; mj = Subscript[m, j]; n0 = Subscript[n, 0];
V0 = Subscript[V, 0];
(*Define our matrix M*)
M = {
{V0 k -\[Omega], 0, k n0, 0, 0},
{0, 0, I(V0 k - \[Omega])me, 0, e},
{0, -\[Omega], 0, k n0, 0},
{0, 0, 0, I mj \[Omega], e},
{1, -1, 0, 0, (I \[Epsilon]0 k)/e}
}; MatrixForm[M]
(*Define and reduce dispersion equation*)
FullSimplify[Det[M] == 0]
(* Define new variables and our equation *)
eq2 = (\[CapitalOmega]^2 - \[Mu]^2)(\[CapitalOmega] - X)^2 - \[CapitalOmega]^2 == 0;
roots[x_] := \[CapitalOmega]/. Solve[eq2/.{\[Mu] -> 1/(Sqrt[1836]), X -> x}, \[CapitalOmega]]
Plot[{Im[roots[x][[1]]], Im[roots[x][[2]]],
Im[roots[x][[3]]],Im[roots[x][[4]]]}, {x, 0, 2},
PlotLegends -> {"Root 1", "Root 2", "Root 3", "Root 4"},
AxesLabel -> {"X","Im(\[CapitalOmega])"}, PlotRange -> All ]
(* --- Define a "Growth Rate" Function --- *)
growthRate[x_?NumericQ] := Max[Im[roots[x]]]
(* --- Use FindRoot to Find Where the Growth Rate is Zero --- *)
Print["--- Precise Instability Boundary ---"];
criticalXResult = FindRoot[
growthRate[x] == 0,
{x, 1.1} (* Our starting guess for the value of x *)
]
x_crit = x /. criticalXResult;
{x->1.1573469675922023`}